OK, Non-Parochial BMDs...
http://www.bmdregisters.co.ukHas several posabilities.
I searched births/baptisms between 1829-1832, figuring she was more likely to put her age up if she didn't have parental concent to migrate.
Also concentrated on London...Kent/Middlesex, figuring her accent would possably give her away to the authorities as well.
And sticking with name Elizabeth, well I would stick with my own name, it's too easy to get caught out if someone yells "Leonie" and I turn to answer if I'd told them me name was Mary.
So, I have 4 Elizabeth DAVIS and 2 Elizabeth DAVIES.
Proplem is, to get full details joining fee of 80 Pounds is required.
So I thought I'd wait and see if anyone has a sub to the site. I'll phone the library, they gave me the link in the first place so if they don't have it the librarian who gave it to me may have it on her home computer. She's a very nice lady, very happy to help and avid genealogist.
Still looking.